Catch the Ultra Beasts
Once you complete the story in Pokemon Sun and Moon, leave your house, and an NPC will approach you. He tells you to meet him at the motel on Route 8. Doing so will initiate the Ultra Beasts quest line, where you aid the International Police in tracking down these powerful beings and capturing them. It’s essentially a hunt for legendary Pokemon, and if you do manage to catch these Ultra Beasts, you’ll be able to use them in battle.
Following the quest line all the way to the end will also give you some closure to the main story, and lets you find out more about the Aether Foundation itself.